SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3185-3211, January 2020.
We consider a Bolza-type optimal control problem for a dynamical system described by a fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1)$. The value of this problem is introduced as a functional in a suitable space of histories of motions. We prove that this functional satisfies the dynamic programming principle. Based on a new notion of coinvariant derivatives of the order $\alpha$, we associate the considered optimal control problem with a Hamilton--Jacobi--Bellman equation. Under certain smoothness assumptions, we establish a connection between the value functional and a solution to this equation. Moreover, we propose a way of constructing optimal feedback controls. The paper concludes with an example.

中文翻译：

---

分数阶系统的动态规划原理和Hamilton-Jacobi-Bellman方程

SIAM控制与优化杂志，第58卷，第6期，第3185-3211页，2020年1月。
我们考虑由分数阶微分方程描述的动力学系统的Bolza型最优控制问题，阶数为\\ alpha \ in（0，1）$。这个问题的价值作为功能引入了运动历史的适当空间中。我们证明该功能满足动态编程原理。基于$ \ alpha $阶协变导数的新概念，我们将考虑的最优控制问题与Hamilton-Jacobi-Bellman方程相关联。在某些平滑度假设下，我们在值函数和该方程式的解决方案之间建立了联系。此外，我们提出了一种构建最佳反馈控制的方法。本文以一个例子结束。